Here's the answer we got back, (with business names removed, of course!)

As I don't work with DNS settings everyday; I hope the options I'm looking for match the right issues:

1. new site at WordPress.com (in progress, not ready to launch yet)

2. we'd like the customer to type in our website (ourdomain.com) in the browser and have the new site from wordpress appear, but with our existing domain name.

3. we want to keep our email on the current domain host's server, as that had sounded like the smoothest way to go.
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Our domain host's answer:
If they say you must use their name servers, then there is no way we can guarantee your email will continue to work. They would have to manage custom DNS entries in their DNS name servers to allow your email to stay on our server, and for it to function properly.

Are they also providing email service as part of the package?

Unfortunately, we can only support your email if your DNS records are set correctly to our mail servers. That has to be done either in our or their name servers one way or another. There has to be a better answer to this, but unfortunately they have to give you the information you require.

You can manage DNS records here with Domain Mapping and pointing name servers to WordPress.COM - you would probably need a MX & A record to make the email work on your old host - i use my old host for email and I had to get some info from him to add to my domain mapping here - end result the email still works - each host is a bit different - I had to get help with the format of the records I needed to add.

 Outgoing mail from personal web client does not reach our website’s email at all. No errors are returned. (using Yahoo, Hotmail, etc. as an example.)

I think I’ve got a syntax error in how I interpreted and made the CNAME change:

When I contacted our email host, here’s their notes:

"It would appear that the CNAME record is incorrect per their instructions. mail.oursite.com should only be mail (the rest of your domain is added automatically per their instructions).
The MX record appears correct, so your email should be getting sent to our server correctly."

To correct our settings, they said:

You need an MX record set with a value of 10 pointing to mail.ouremailhost.com
You need a CNAME record pointing mail.oursite.com to mail. ouremailhost.com

Here’s what I’ve currently entered in the custom DNS records for our site: